Problem 3

Explain why when different powers of the same quantity are multiplied together their exponents are added.

Solution. Consider $x^m$ and $x^n$. By definition,

$x^m=x\cdot x\cdots x$ (multiply $x$ by itself $m$ times)

and

$x^n=x\cdot x\cdots x$(multiply $x$ by itself $n$ times)

Their product is $x$ multiplied by itself $m$ times multiplied by $x$ multiplied by itself $n$ . Another words, $x$ multiplied by itself $m+n$ times

$\Rightarrow$ $x^m\cdot x^n=x^{m+n}$

An alternative explanation uses logarithms:

$\log \left(x^m\right)=m\log \left(x\right)$

and

$\log \left(x^n\right)=n\log \left(x\right)$

$\Rightarrow \log (x^m\cdot x^n)=(m+n)\log \left(x\right)$

and exponentiating both sides gives the same result.

Answer: $x^m\cdot x^n=x^{m+n}$